arXiv subjects

Elias Döhrer

Publications and source records attributed to Elias Döhrer.

4 recordsLinked to original sources

arXiv2026

A Lusin theorem for nonlocal gradients

We extend the celebrated result of Alberti, stating that Borel vector fields coincide with gradients of $C^1$-functions outside of a set of arbitrary small measure. We prove that a similar statement holds true in the setting of fractional gradients and $C^{0,s}-functions. Furthermore, we establish analogous results for general nonlocal gradients with appropriate function spaces. Both statements are proven by means of the translation method, demonstrating its use for the purposes of geometric measure theory. In particular, this article illustrates how the translation method transfers rigidity phenomena from the classical gradient to a broad class of nonlocal gradients.

math.CA↗

arXiv2026

A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energies

In this paper, we prove a Fenchel theorem for Gauss maps by providing sharp lower bounds for the path length of Gauss maps of an embedding. By combining the Fenchel-type theorem with various techniques from the field of geometric analysis, we show that circles minimize most generalized tangent-point energies. Furthermore, we prove that disks minimize all fractional Willmore energies among the class of convex planar sets.

math.CA↗

arXiv2025

On a Complete Riemannian Metric on the Space of Embedded Curves

We propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of regularity $H^s$, $s\in(3/2,2)$. We highlight its close relationship to the (generalized) tangent-point energies and employ it to show that this metric is complete in the following senses: (i) bounded sets are relatively compact with respect to the weak $H^s$ topology; (ii) every Cauchy sequence with respect to the induced geodesic distance converges; (iii) solutions of the geodesic initial-value problem exist for all times; and (iv) there are length-minimizing geodesics between every pair of curves in the same path component (i.e., in the same knot class). As a by-product, we show $C^\infty$-smoothness of the tangent-point energies in the Hilbert case.

math.DG↗

arXiv2025

Convergence of gradient flows on knotted curves

We prove full convergence of gradient-flows of the arc-length restricted tangent point energies in the Hilbert-case towards critical points. This is done through a Łojasiewicz-Simon gradient inequality for these energies. In order to do so, we prove, that the tangent-point energies are anlytic on the manifold of immersed embeddings and that their Hessian is Fredholm with index zero on the manifold of arc-length parametrized curves. As a by-product, we also show that the metric on the manifold of embedded immersed curves, defined by the first author, is analytic.

math.CA↗